This presentation concerns recently developed methods to solve elliptic problems over infinite domains. Solving elliptic PDEs over infinite domains is relevant in various applications in mathematical sciences. A naive approach to solve these problems is to truncate the infinite domain and enforce artificial boundary conditions on the boundary of the domain. This approach, however, results in large errors due to the poor decay of the Green's function associated with elliptic operators. In this talk, we present a recently developed approach for addressing this problem in two separate contexts: homogenization problems and micro-magnetism, whereby two different theoretical set-ups emerge.